Abstract

An approximate expression of a Bessel-Gaussian beam (BGB) with desired topological charge is introduced using a coherence superposition of decentered Gaussian beams (dGBs). And based on such an expression and the extended Huygens-Fresnel principle, the propagation properties of BGBs traveling in turbulent atmosphere are explored. An analytical expression of the average intensity of a BGB with phase singularity propagating through turbulent atmosphere is obtained and analyzed numerically. It is found that intensity profiles of BGBs experienced successive variations and the phase singularity rapidly fades away during propagating in turbulent atmosphere.

© 2008 Optical Society of America

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  1. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, (Bellingham, Washington: SPIE Optical Engineering Press; 1998).
  2. Y. Cai, "Propagation of various flat-topped beams in a turbulent atmosphere," J. Opt. A 8, 537-545 (2006).
    [CrossRef]
  3. E. T. Eyyuboglu, C. Arpali, and Y. Baykal, "Flat topped beams and their characteristics in turbulent media," Opt. Express 14, 4196-4207 (2006).
    [CrossRef] [PubMed]
  4. H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405(2006).
    [CrossRef]
  5. J. Wu and A. D. Boradman, "Coherence length of a Gaussian-Schell beam and atmosphere turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
    [CrossRef]
  6. J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 38, 671-684 (1991).
  7. G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [CrossRef]
  8. A. Dogariu and S. Amarande, "Propagation of partially coherent beams: turbulence-induced degradation," Opt. Lett. 28, 10-12 (2003).
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  9. T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003).
    [CrossRef]
  10. Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 0411171-3 (2006).
    [CrossRef]
  11. H. T. Eyyuboglu, "Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence," Opt. Laser Technol. 40, 156-166 (2008).
    [CrossRef]
  12. G. Gbur and O. Korotkova, "Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence," J. Opt. Soc. Am. A 24, 745-752 (2007).
    [CrossRef]
  13. M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42. 219-276 (2001).
    [CrossRef]
  14. J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
    [CrossRef]
  15. M. V. Berry, "Singularities in waves and rays," in Les Houches Session XXV-Physics of Defects, R. Balian, M. Kleman, and J.-P. Poirier, eds., (North-Holland; 1981), p. 453-543.
  16. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
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  19. H. T. Eyyuboglu, "Propagation of higher order Bessel-Gaussian beams in turbulence," Appl. Phys. B 88, 259-265 (2007).
    [CrossRef]
  20. P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, "Effect of phase fluctuations on propagation of the vortex beams," Atmos. Oceanic Opt. 19, 924-927 (2006).
  21. V. P. Aksenov and Ch. E. Pogutsa, "Fluctuations of the orbital angular momentum of a laser beam, carrying an optical vortex, in the turbulent atmosphere," Quantum Electron. 38, 343-348 (2008).
    [CrossRef]
  22. G. Gbur and R. K. Tyson, "Vortex beam propagation through atmospheric turbulence and topological charge conservation," J. Opt. Soc. Am. A. 25, 225-229 (2008).
    [CrossRef]
  23. B. S. Chen, Z. Y. Chen, and J. X. Pu, "Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere," Opt. Laser Technol. 40, 820-827 (2008).
    [CrossRef]
  24. T. Wang, J. X. Pu, and Z. Y. Chen, "Propagation of partially coherent vortex beams in a turbulent atmosphere," Opt. Eng. 47, 036002 (2008).
    [CrossRef]
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  27. J. Yin, W. Gao, and Y. Zhu, "Generation of dark hollow beams and their applications," in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2003), Vol. 44, p.119-204.
    [CrossRef]
  28. Y. J. Cai, Z. Y. Wang, and Q. Lin, "An alternative theoretical model for an anomalous hollow beam," Opt. Express 16, 15254-15267 (2008).
    [CrossRef] [PubMed]
  29. K. C. Zhu, H. Q. Tang, and Y. Y. Gao, "A new set of flattened light beams," J. Opt. A: Pure Appl. Opt. 4, 33-36 (2002).
    [CrossRef]
  30. K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik 113, 222-226 (2002).
    [CrossRef]
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2008 (6)

H. T. Eyyuboglu, "Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence," Opt. Laser Technol. 40, 156-166 (2008).
[CrossRef]

V. P. Aksenov and Ch. E. Pogutsa, "Fluctuations of the orbital angular momentum of a laser beam, carrying an optical vortex, in the turbulent atmosphere," Quantum Electron. 38, 343-348 (2008).
[CrossRef]

G. Gbur and R. K. Tyson, "Vortex beam propagation through atmospheric turbulence and topological charge conservation," J. Opt. Soc. Am. A. 25, 225-229 (2008).
[CrossRef]

B. S. Chen, Z. Y. Chen, and J. X. Pu, "Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere," Opt. Laser Technol. 40, 820-827 (2008).
[CrossRef]

T. Wang, J. X. Pu, and Z. Y. Chen, "Propagation of partially coherent vortex beams in a turbulent atmosphere," Opt. Eng. 47, 036002 (2008).
[CrossRef]

Y. J. Cai, Z. Y. Wang, and Q. Lin, "An alternative theoretical model for an anomalous hollow beam," Opt. Express 16, 15254-15267 (2008).
[CrossRef] [PubMed]

2007 (2)

2006 (5)

P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, "Effect of phase fluctuations on propagation of the vortex beams," Atmos. Oceanic Opt. 19, 924-927 (2006).

Y. Cai, "Propagation of various flat-topped beams in a turbulent atmosphere," J. Opt. A 8, 537-545 (2006).
[CrossRef]

E. T. Eyyuboglu, C. Arpali, and Y. Baykal, "Flat topped beams and their characteristics in turbulent media," Opt. Express 14, 4196-4207 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405(2006).
[CrossRef]

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 0411171-3 (2006).
[CrossRef]

2003 (2)

2002 (3)

G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[CrossRef]

K. C. Zhu, H. Q. Tang, and Y. Y. Gao, "A new set of flattened light beams," J. Opt. A: Pure Appl. Opt. 4, 33-36 (2002).
[CrossRef]

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik 113, 222-226 (2002).
[CrossRef]

2001 (1)

M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42. 219-276 (2001).
[CrossRef]

2000 (2)

1998 (1)

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

1991 (2)

J. Wu and A. D. Boradman, "Coherence length of a Gaussian-Schell beam and atmosphere turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 38, 671-684 (1991).

1987 (1)

1977 (1)

1974 (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Aksenov, V. P.

V. P. Aksenov and Ch. E. Pogutsa, "Fluctuations of the orbital angular momentum of a laser beam, carrying an optical vortex, in the turbulent atmosphere," Quantum Electron. 38, 343-348 (2008).
[CrossRef]

Amarande, S.

Arpali, C.

Baykal, Y.

E. T. Eyyuboglu, C. Arpali, and Y. Baykal, "Flat topped beams and their characteristics in turbulent media," Opt. Express 14, 4196-4207 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405(2006).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Boradman, A. D.

J. Wu and A. D. Boradman, "Coherence length of a Gaussian-Schell beam and atmosphere turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

Cai, Y.

Y. Cai, "Propagation of various flat-topped beams in a turbulent atmosphere," J. Opt. A 8, 537-545 (2006).
[CrossRef]

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 0411171-3 (2006).
[CrossRef]

Cai, Y. J.

Capjack, C. E.

Chen, B. S.

B. S. Chen, Z. Y. Chen, and J. X. Pu, "Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere," Opt. Laser Technol. 40, 820-827 (2008).
[CrossRef]

Chen, Z. Y.

B. S. Chen, Z. Y. Chen, and J. X. Pu, "Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere," Opt. Laser Technol. 40, 820-827 (2008).
[CrossRef]

T. Wang, J. X. Pu, and Z. Y. Chen, "Propagation of partially coherent vortex beams in a turbulent atmosphere," Opt. Eng. 47, 036002 (2008).
[CrossRef]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Dogariu, A.

Durnin, J.

Eyyuboglu, E. T.

Eyyuboglu, H. T.

H. T. Eyyuboglu, "Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence," Opt. Laser Technol. 40, 156-166 (2008).
[CrossRef]

H. T. Eyyuboglu, "Propagation of higher order Bessel-Gaussian beams in turbulence," Appl. Phys. B 88, 259-265 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405(2006).
[CrossRef]

Gao, Y. Y.

K. C. Zhu, H. Q. Tang, and Y. Y. Gao, "A new set of flattened light beams," J. Opt. A: Pure Appl. Opt. 4, 33-36 (2002).
[CrossRef]

Gbur, G.

He, S.

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 0411171-3 (2006).
[CrossRef]

Ishimaru, A.

Konyaev, P. A.

P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, "Effect of phase fluctuations on propagation of the vortex beams," Atmos. Oceanic Opt. 19, 924-927 (2006).

Korotkova, O.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Lin, Q.

Liu, T. N.

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik 113, 222-226 (2002).
[CrossRef]

Lu, B.

Lukin, V. P.

P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, "Effect of phase fluctuations on propagation of the vortex beams," Atmos. Oceanic Opt. 19, 924-927 (2006).

Ma, H.

Nye, J. F.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Pogutsa, Ch. E.

V. P. Aksenov and Ch. E. Pogutsa, "Fluctuations of the orbital angular momentum of a laser beam, carrying an optical vortex, in the turbulent atmosphere," Quantum Electron. 38, 343-348 (2008).
[CrossRef]

Pu, J. X.

T. Wang, J. X. Pu, and Z. Y. Chen, "Propagation of partially coherent vortex beams in a turbulent atmosphere," Opt. Eng. 47, 036002 (2008).
[CrossRef]

B. S. Chen, Z. Y. Chen, and J. X. Pu, "Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere," Opt. Laser Technol. 40, 820-827 (2008).
[CrossRef]

Seguin, H. J. J.

Sennikov, V. A.

P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, "Effect of phase fluctuations on propagation of the vortex beams," Atmos. Oceanic Opt. 19, 924-927 (2006).

Sermutlu, E.

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405(2006).
[CrossRef]

Shirai, T.

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42. 219-276 (2001).
[CrossRef]

Strohschein, J. D.

Tang, H. Q.

K. C. Zhu, H. Q. Tang, and Y. Y. Gao, "A new set of flattened light beams," J. Opt. A: Pure Appl. Opt. 4, 33-36 (2002).
[CrossRef]

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik 113, 222-226 (2002).
[CrossRef]

Tovar, A. A.

Tyson, R. K.

G. Gbur and R. K. Tyson, "Vortex beam propagation through atmospheric turbulence and topological charge conservation," J. Opt. Soc. Am. A. 25, 225-229 (2008).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42. 219-276 (2001).
[CrossRef]

Wang, T.

T. Wang, J. X. Pu, and Z. Y. Chen, "Propagation of partially coherent vortex beams in a turbulent atmosphere," Opt. Eng. 47, 036002 (2008).
[CrossRef]

Wang, X. W.

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik 113, 222-226 (2002).
[CrossRef]

Wang, Z. Y.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Wolf, E.

Wu, J.

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 38, 671-684 (1991).

J. Wu and A. D. Boradman, "Coherence length of a Gaussian-Schell beam and atmosphere turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

Zhu, K. C.

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik 113, 222-226 (2002).
[CrossRef]

K. C. Zhu, H. Q. Tang, and Y. Y. Gao, "A new set of flattened light beams," J. Opt. A: Pure Appl. Opt. 4, 33-36 (2002).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

H. T. Eyyuboglu, "Propagation of higher order Bessel-Gaussian beams in turbulence," Appl. Phys. B 88, 259-265 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 0411171-3 (2006).
[CrossRef]

Atmos. Oceanic Opt. (1)

P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, "Effect of phase fluctuations on propagation of the vortex beams," Atmos. Oceanic Opt. 19, 924-927 (2006).

J. Mod. Opt. (2)

J. Wu and A. D. Boradman, "Coherence length of a Gaussian-Schell beam and atmosphere turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 38, 671-684 (1991).

J. Opt. A (1)

Y. Cai, "Propagation of various flat-topped beams in a turbulent atmosphere," J. Opt. A 8, 537-545 (2006).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

K. C. Zhu, H. Q. Tang, and Y. Y. Gao, "A new set of flattened light beams," J. Opt. A: Pure Appl. Opt. 4, 33-36 (2002).
[CrossRef]

J. Opt. Soc. Am. A (6)

J. Opt. Soc. Am. A. (1)

G. Gbur and R. K. Tyson, "Vortex beam propagation through atmospheric turbulence and topological charge conservation," J. Opt. Soc. Am. A. 25, 225-229 (2008).
[CrossRef]

Opt. Commun. (2)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405(2006).
[CrossRef]

Opt. Eng. (1)

T. Wang, J. X. Pu, and Z. Y. Chen, "Propagation of partially coherent vortex beams in a turbulent atmosphere," Opt. Eng. 47, 036002 (2008).
[CrossRef]

Opt. Express (2)

Opt. Laser Technol. (2)

H. T. Eyyuboglu, "Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence," Opt. Laser Technol. 40, 156-166 (2008).
[CrossRef]

B. S. Chen, Z. Y. Chen, and J. X. Pu, "Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere," Opt. Laser Technol. 40, 820-827 (2008).
[CrossRef]

Opt. Lett. (1)

Optik (1)

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik 113, 222-226 (2002).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Prog. Opt. (1)

M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42. 219-276 (2001).
[CrossRef]

Quantum Electron. (1)

V. P. Aksenov and Ch. E. Pogutsa, "Fluctuations of the orbital angular momentum of a laser beam, carrying an optical vortex, in the turbulent atmosphere," Quantum Electron. 38, 343-348 (2008).
[CrossRef]

Other (4)

J. Yin, W. Gao, and Y. Zhu, "Generation of dark hollow beams and their applications," in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2003), Vol. 44, p.119-204.
[CrossRef]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1980), p. 718.

M. V. Berry, "Singularities in waves and rays," in Les Houches Session XXV-Physics of Defects, R. Balian, M. Kleman, and J.-P. Poirier, eds., (North-Holland; 1981), p. 453-543.

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, (Bellingham, Washington: SPIE Optical Engineering Press; 1998).

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Figures (3)

Fig. 1.
Fig. 1.

Intensity profile (Upper row) and corresponding phase distributions (Lower row) for M=12, R=4w 0 and several different charges n=1, 2, 3, and 4.

Fig. 2.
Fig. 2.

Cross line (y=0) of the normalized intensity distribution at several selected distances in a turbulent atmosphere for charges n=2 or 3, C 2 n =10-14 m -2/3 and w 0=0.01m.

Fig. 3.
Fig. 3.

Cross line (y=0) of the normalized intensity distribution at z=0.4km (a) for w 0=0.01m and C 2 n :10-15 (Dotted line), 10-14 (Solid line) and 10-13 m -2/3 (Dash-dot line) and w 0=0.01m; (b) for C 2 n =10-14 m -2/3 and w 0=0.015m(Dotted line), 0.01m(Solid line) and 0.007m (Dash-dot line), while n=2.

Tables (1)

Tables Icon

Table 1 Dependence of a n on the charge n for R=4w 0 or 6w 0.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

E ( x , y , 0 ) = m = 0 M 1 E m ( x , y , 0 )
E m ( x , y , 0 ) = exp [ ( x i 2 R cos θ m ) 2 + ( y i 2 R sin θ m ) 2 w 0 2 + i φ m ]
E ( ρ , φ , 0 ) = exp ( ρ 2 1 4 R 2 w 0 2 ) p = i p J p ( R ρ w 0 2 ) exp ( i p φ ) m = 0 M 1 exp [ im ( n p ) α 0 ]
m = 0 M 1 exp [ im ( n p ) α 0 ] = M δ p n , Mq ( q = 0 , ± 1 , )
E ( ρ , φ , 0 ) = M exp ( ρ 2 1 4 R 2 w 0 2 ) q = i n + Mq J n + Mq ( R ρ w 0 2 ) exp [ i ( n + Mq ) φ ]
a n + Mq = 0 exp ( 2 ρ 2 ) J n + Mq 2 ( R ρ w 0 ) ρ d ρ q = 0 exp ( 2 ρ 2 ) J n + Mq 2 ( R ρ w 0 ) ρ d ρ
= I n + Mq ( R 2 4 w 0 2 ) q = I n + Mq ( R 2 4 w 0 2 )
E ( x , y , 0 ) = J n ( R ρ w 0 2 ) exp ( ρ 2 w 0 2 + in φ )
1 M i n exp ( R 2 4 w 0 2 ) m = 0 M 1 exp [ ( x i 2 R cos θ m ) 2 + ( y i 2 R sin θ m ) 2 w 0 2 + i φ m ]
I ( x , y , z ) = k 2 ( 2 π z ) 2 E ( p , q , 0 ) E * ( ξ , η , 0 ) exp [ ψ ( p , q , x , y ) + ψ * ( ξ , η , x , y ) ] m
· exp ( ik 2 z [ ( x p ) 2 + ( y q ) 2 ( x ξ ) 2 ( y η ) 2 ] ) dpdqd ξ d η
exp [ ψ ( p , q , x , y ) + ψ * ( ξ , η , , x , y ) ] = exp [ 0.5 D ψ ( p ξ , q η ) ] = exp [ ( p ξ ) 2 + ( q η ) 2 ρ 0 2 ]
I ( x , y , z ) = k 2 ( 2 π z ) 2 E ( p , q , 0 ) E * ( ξ , η , 0 )
· exp ( ik 2 z [ ( x p ) 2 + ( y q ) 2 ( x ξ ) 2 ( y η ) 2 ] 1 ρ 0 2 [ ( p ξ ) 2 + ( q η ) 2 ] ) dpdqd ξ d η
I ( x , y , z ) = N w 2 M 2 Ω exp { R 2 ( 1 + w 0 2 ρ 0 2 ) 2 Ω w 0 2 2 N w 2 Ω x 2 + y 2 w 0 2 } m , J M 1 exp [ i ( φ m φ l ) + R 2 cos ( θ m θ l ) 2 Ω ρ 0 2 ]
· exp { N w ( 1 + i N w ) R Ω w 0 2 ( x cos θ m + y sin θ m ) + N w ( 1 i N w ) R Ω w 0 2 ( x cos θ l + y sin θ l ) }

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